sl<sub>2</sub> is the fundamental example of a finite dimensional simple Lie algebra.  It forms the basis of the Cartan-Killing classification of complex semisimple Lie algebras, since all of the others can be made by gluing (in some sense) copies of sl<sub>2</sub> together.  For the same reason, SU(2) is the fundamental example of a nonabelian compact Lie group.